quantum dilogarithm

In mathematics, the quantum dilogarithm is a special function defined by the formula

:$$

\phi(x)\equiv(x;q)_\infty=\prod_{n=0}^\infty (1-xq^n),\quad |q|<1

It is the same as the q-exponential function $e\_q(x)$.

Let $u,v$ be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation $uv=qvu$. Then, the quantum dilogarithm satisfies Schützenberger's identity

:$\backslash phi(u)\; \backslash phi(v)=\backslash phi(u\; +\; v),$

Faddeev-Volkov's identity

:$\backslash phi(v)\; \backslash phi(u)=\backslash phi(u\; +v\; -vu),$

and Faddeev-Kashaev's identity

:$\backslash phi(v)\backslash phi(u)=\backslash phi(u)\backslash phi(-vu)\backslash phi(v).$

The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity.

Faddeev's quantum dilogarithm $\backslash Phi\_b(w)$ is defined by the following formula:

: $\backslash Phi\_b(z)=\backslash exp$

\left(

\frac{1}{4}\int_C

\frac{e^{-2i zw }}

{\sinh (wb) \sinh (w/b) }

\frac{dw}{w}

\right),

where the contour of integration $C$ goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz:

: $$

\Phi_b(x)=\exp\left(\frac{i}{2\pi}\int_{\mathbb R}\frac{\log(1+e^{tb^2+2\pi b x})}{1+e^{t}}\,dt\right).

Ludvig Faddeev discovered the quantum pentagon identity:

: $\backslash Phi\_b(\backslash hat\; p)\backslash Phi\_b(\backslash hat\; q)$

=

\Phi_b(\hat q)

\Phi_b(\hat p+ \hat q)

\Phi_b(\hat p),

where $\backslash hat\; p$ and $\backslash hat\; q$ are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

:$[\backslash hat\; p,\backslash hat\; q]=\backslash frac1\{2\backslash pi\; i\}$

and the inversion relation

: $\backslash Phi\_b(x)\backslash Phi\_b(-x)=\backslash Phi\_b(0)^2\; e^\{\backslash pi\; ix^2\},\backslash quad\; \backslash Phi\_b(0)=e^\{\backslash frac\{\backslash pi\; i\}\{24\}\backslash left(b^2+b^\{-2\}\backslash right)\}.$

The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.

The precise relationship between the q-exponential and $\backslash Phi\_b$ is expressed by the equality

:$\backslash Phi\_b(z)=\backslash frac\{E\_\{e^\{2\backslash pi\; ib^2\}\}(-e^\{\backslash pi\; ib^2+2\backslash pi\; zb\})\}\{E\_\{e^\{-2\backslash pi\; i/b^2\}\}(-e^\{-\backslash pi\; i/b^2+2\backslash pi\; z/b\})\},$

valid for $\backslash operatorname\{Im\}\; b^2>0$.